g10-05-04 · Core topics
Keep every restriction after simplification
Does cancelling x−1 restore the missing input x=1?
1 MIN · PREDICT ON PAPER
Predict whether the original and simplified formulas both have value 2 at x=1.
Discuss or write first. No typed answer is required to continue.
Overlaid masks: 2/2. Current intersection accepts x = 0.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
For √(x+1)/(x−2), take x ≥ −1 and x ≠ 2 together. For (x²−1)/(x−1), cancellation gives x+1 only on x ≠ 1. The original graph retains its hole.
Conditions: The original formula fixes the domain. Finite output sets are not filled in between their elements.
Mathematical justification: Factor x²−1=(x−1)(x+1); division by x−1 is equivalent only if x≠1. Separately, x² on [−2,1] attains every y∈[0,4] via x=−√y.
1 MIN · EXIT CHECK
Is (1,2) on the original graph of (x²−1)/(x−1)?
Check after answering
No. At x=1 the original denominator is zero.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Let students overlay both restriction masks, then compare arithmetic and domain membership at the hole.
Watch for: A simpler expression automatically enlarges the original function domain.
Challenge: Compare the ranges of x² on {−2,−1,0,1} and on [−2,1]. Why is only one a full interval?
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
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